On using different error measures for fuzzy linear regression analysis

Transcription

1 On using different error measures for fuzzy linear regression analysis Duygu İÇEN Hacettepe University Department of Statistics Ankara /TURKEY 2013

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4 Presentation Plan Introduction Some Definitions Fuzzy linear regression with Monte Carlo method The simulation study Application Conclusion 4

5 Introduction Regression analysis is a statistical tool used to figure out the mathematical relation between two or more quantitative variables. There are many types of regression techniques in the literature. Most of these approaches are rather restrictive, and their application to real life problems requires various assumptions. Therefore new techniques have been proposed to relax some of these assumptions. All of these authors try to find analytical solutions for the estimators of regression parameters. Buckley and Abdalla [1, 2, 3] are the first practitioners of MC method into fuzzy linear regression analysis 5

6 Some Definitions 6

7 Some Definitions Fuzzy Number A ~ a,a, 1 2 a3 7

8 Some Definitions 8

9 Some Definitions 9

10 Some Definitions ** ** Omar A. AbuAarqob, Nabil T. Shawagfeh and Omar A. AbuGhneim, (2008) Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension, International Mathematical Forum, 3,, no. 16,

11 Some Definitions 11

12 Fuzzy linear regression with Monte Carlo method 12

13 Fuzzy linear regression with Monte Carlo method 13

14 Fuzzy linear regression with Monte Carlo method 14

15 Fuzzy linear regression with Monte Carlo method 15

16 The simulation study 16

17 The simulation study 17

18 The simulation study 18

19 The simulation study 19

20 The simulation study 20

21 The simulation study 21

22 The simulation study 22

23 Application Fuzzy Output x 1 x 2 x 3 (2.27/5.83/9.39) (0.33/0.85/1.37) (5.43/13.93/22.43) (1.56/4.00/6.44) (0.64/1.65/2.66) (0.62/1.58/2.54) (3.19/8.18/13.17) (0.72/1.85/2.98)

24 Application 24

25 Application 25

26 Application Fuzzy Output X 1 X 2 (55.4/61.6/64.7) (5.7/6.0/6.9) (5.4/6.3/7.1) (50.5/53.2/58.5) (4.0/4.4/5.1) (4.7/5.5/5.8) (55.7/65.5/75.3) (8.6/9.1/9.8) (3.4/3.6/4.0) (61.7/64.9/74.7) (6.9/8.1/9.3) (5.0/5.8/6.7) (69.1/72.7/80.0) (8.7/9.4/11.2) (6.5/6.8/7.1) (49.6/52.2/57.4) (4.6/4.8/5.5) (6.7/7.9/8.7) (47.7/50.2/55.2) (7.2/7.6/8.7) (4.0/4.2/4.8) (41.8/44.0/48.4) (4.2/4.4/4.8) (5.4/6.0/6.3) (45.7/53.8/61.9) (8.2/9.1/10.0) (2.7/2.8/3.2) (45.4/53.5/58.9) (6.0/6.7/7.4) (5.7/6.7/7.7) 26

27 The simulation study 27

28 Conclusion In this study, we use different error measures to find the parameter estimates of fuzzy linear regression models with MC method. A simulation study is conducted to compare the estimation performances of the error measures we mentioned. We showed that only two error measures (E1 and E2) are not enough for estimating the parameters of fuzzy linear regression models. We also estimate the parameters with considering five different intervals from where they come. it is possible to say that best error measures to estimate fuzzy/crisp parameters of fuzzy regression models are not only E1 and E2 but also MSEe. Furthermore the worst error measure is MPEe for estimating the parameters of fuzzy regression models. 28

29 Future Works Considering more than one way to get the absolute value of the triangular fuzzy number, it is possible to apply different methods in MC method in fuzzy linear regression analysis. Extension of the proposed method for different type of fuzzy regression models, such as nonparametric fuzzy regression or fuzzy nonlinear regression, is a potential area for the future work. The most important thing for fuzzy linear regression model is deciding the intervals about the parameters. New methods can be applied to choose the convenient intervals. For example expert systems or fuzzy expert systems. 29

30 [1] Abdalla A, Buckley JJ (2008) Monte Carlo methods in fuzzy linear regression II. Soft Comput, 12: [2] Abdalla A, Buckley JJ (2008) Monte Carlo methods in fuzzy nonlinear regression. New Mathematics and Natural Computation, Vol.4,No.2, [3] Abdalla A, Buckley JJ (2007) Monte Carlo methods in fuzzy linear regression. Soft Comput, 11: [4] AbuAarqob OA,Shawagfeh NT, AbuGhneim OA (2008) Functions Defined on Fuzzy Real Numbers According to Zadeh's Extension. International Mathematical Forum, 3, no. 16, [5] Alefeldt G,Claudio D, (1998) The Basic Properties of Interval Arithmetic; its Software Realizations and Some Applications. Computers and Structures, 67, 3-8 [6] Bardossy A, Hagaman R, Duckstein L and Bogardi I (1992) Fuzzy least-squares regression: theory and applications. In J. Kacprzyk and M. Fedrizzi, editors. Fuzzy Regression Analysis, pages Physica- Verlag, Heidelberg [7] Buckley JJ, Jowers LJ (2008) Monte Carlo Methods in Fuzzy Optimization. Springer-Verlag,Berlin,Heidelberg [8] Choi HS, Buckley JJ (2008) Fuzzy regression using least absolute deviation estimators. Soft Comput, 12, [9] Diamond P (1987) Least squares tting of several fuzzy variables. In Proc of Second IFSA Congress, p.20-25, IFSA, Tokyo [10] Diamond P, Korner R (1997) Extended fuzzy linear models and least squares estimates. Comput Math Appl 33:15-32 [11] Dubois D, Prade H (1978) Operations with fuzzy numbers. Int J Syst Sci, 9(6): [12] Hong DH, Song J and Do HY,(2001) Fuzzy least-squares linear regression analysis using shape preserving operations. Inform. Sci., 138, [13] Kao C, Chyu C (2002) A fuzzy linear regression model with better explanatory power. Fuzzy Sets Syst 126: [14] Kim B, Bishu RR (1998) Evaluation of fuzzy linear regression models by comparing membership functions. Fuzzy Sets Syst, 100: [15] Luczynski W and Matloka M (1995) Fuzzy regression models and their applications. J. Fuzzy Math., 3: [16] Nather W and Korner R (1998) Linear regression with random fuzzy numbers. Uncertainty Analysis in Engineering and Sciences, p Kluwer, Boston [17] Peters G (1994) Fuzzy linear regression with fuzzy intervals. Fuzzy Sets Syst., 63:45-55 [18] Savic DA, Pedryzc W (1991) Evaluation of fuzzy linear regression models. Fuzzy Sets Syst. 39:51-63 [19] Taheri SM (2003) Trends in Fuzzy Statistics. Austrian Journal of Statistics, 32:3, [20] Tanaka H (1987) Fuzzy data analysis by possibilistic linear regression models. Fuzzy Sets Syst 24: [21] Tanaka H, Ishibuchi H and Yoshikawa S (1995) Exponential possibility regression analysis. Fuzzy Sets Syst., 69: [22] Tanaka H, Uejima S, Asai K (1982) Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybernet.12, [23] Tanaka H, Uejima S, Asai K (1980) Fuzzy linear regression model. In:International Congress on Applied Systems and Cybernetics, Vol.VI. Acapulco, Mexico, VI,pp [24] Yen K.K., Ghoshray S. and Roig G (1999) A linear regression model using triangular fuzzy number coecients. Fuzzy Sets Syst., 106: [25] Zadeh LA (1965) Fuzzy Sets. Inform Control, 8,

31 On using different error measures for fuzzy linear regression analysis Duygu İÇEN Hacettepe University Department of Statistics Ankara /TURKEY 2013